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Full text of "Treatise On Analysis Vol-Ii"

CHAPTER XIV

INTEGRATION IN LOCALLY COMPACT GROUPS

Haar measure and convolution on arbitrary locally compact groups have
become indispensable tools for the modern analyst, as they were in classical
analysis on the real line and in finite-dimensional Euclidean spaces. Together
with convolution of distributions, which generalizes convolution of measures
and which we shall introduce in Chapter XVII, they are the fundamental
notions in harmonic analysis (Chapter XXII) and the theory of linear repre-
sentations of compact groups (Chapter XXI).

We have again followed, though in less detail, the exposition of N. Bourbaki
[22]. In fact, since in this treatise practically the only locally compact groups
we shall consider will be Lie groups (Chapters XVI, XIX, and XXI), for which
there is a much simpler proof of the existence of a Haar measure, we could
have restricted ourselves entirely to Lie groups. Nevertheless, it seemed
worthwhile to bring out the fact that the theory of integration on a locally
compact group is entirely independent of any differential structure; and
the totally discontinuous locally compact groups have ceased to be mere
curiosities since the advent of/?-adic and "adelic" groups in the theory of
numbers [36].

Throughout this chapter, the phrase "locally compact group" mil mean
"separable metrizable locally compact group"

1. EXISTENCE AND UNIQUENESS OF HAAR MEASURE

Let G be a (separable, metrizable) locally compact group. If/is any
mapping of G into a set E and if s is any element of G, we define the left and
right translations y(s)/and 5(s)/of/by s, which are mappings of G into E, by

242such that \g(y)\ > a. If